Particle production in these interactions also depends upon the assumed HIM. Using CRMC v1.6.0 [1]}, we built interaction matrices for the EPOS-LHC [2]} and Sibyll2.3c [3]} HIMs. Simulating the collision between a CR of nucleus \(A\)  and energy \(E_j\)  with a proton at rest, we generated an interaction matrix \(I_{ij}^{sA,m}\)  by tabulating all the secondaries of type \(s\)  (e.g. \(\nu \) , \(\pi \) , \(\gamma \) , \(A^{\prime }\) ) and energy \(E_i\)  predicted by HIM \(m\) . Pions and neutrons were treated as stable for this purpose, in order to treat these particles in a consistent way with those produced in photohadronic interactions (as described in [4]}), and all other particles were decayed before being tabulated. The final interaction matrices give the average number of secondaries of energy \(E_i\)  produced in a collision. With the interaction matrices in hand, one can easily compute the total number of secondaries \(s\)  produced in logarithmic interval \([\lg {E_i}, \lg {E_i}+\mathrm {d}\lg {E_i})\)  by a flux of CRs as
